Quick break from prepping and grading final exams.
My future elementary teachers always struggle to name the denominator when they need to find of .
They draw the picture.
They know that the numerator needs to be 3. And then they argue about whether the denominator should be 12 or 16.
I struggle every year to get 8 on the table as an acceptable answer. I usually end up being a voice of authority for 8, and we discuss what the whole is if you use 12 or 16 as the denominator.
My students don’t like 8 because that means the answer is of one square, but the pieces come from different squares.
This year, I had an insight that helped a lot. The question was this:
What are some situations in life when you get two same-sized parts of distinct wholes?
I opened the class session following our usual denominator debate with this question and it helped us to focus on the issue at hand.
After a few false starts (i.e. examples that didn’t really exemplify what we were after), we settled on this scenario.
When you buy a 75¢ pop from a vending machine, by inserting a dollar you get back a quarter. Do it again and now you have two quarters. Each quarter came from a different dollar, but they are still quarters. Each is one-fourth of a dollar and together they are half a dollar (even though collectively, they are one-fourth of the money you started with).
Back to the squares and we had a frame of reference for eighths.
I have been teaching this course for 8 years.